2.092/2.093
FINITE ELEMENT ANALYSIS OF SOLIDS AND FLUIDS I
FALL 2009
Homework 8
Instructor:
TA:
Prof. K. J. Bathe
Seounghyun Ham
Assigned: Session 23
Due: Session 25
Problem 1 (20 points):
Consider Problem 1 of Homework 7.
a) Calculate the static correction to the analysis performed in Homework 7.
b) Compare by plots the solutions obtained with (i) using one mode plus the static correction
and (ii) using two modes, and discuss your results.
Problem 2 (10 points):
Establish a Rayleigh damping matrix C for the system of Problem 1 of Homework 7, which
gives modal damping parameters, ξ1=0.02 and ξ2=0.10.
Problem 3 (20 points):
Consider the generalized eigenvalue problem
⎡ 4 −1 0 ⎤
⎡2 0 0⎤
⎢ −1 3 −1⎥ φ = λ ⎢ 0 2 1 ⎥ φ
⎢
⎥
⎢
⎥
⎢⎣ 0 −1 4 ⎥⎦
⎢⎣ 0 1 2 ⎥⎦
a) Calculate the eigenvalues and eigenvectors and show explicitly that these vectors are Mand K- orthogonal.
b) Find (any) two vectors that are M- and K- orthogonal but are not eigenvectors.
1
Problem 4 (20 points):
Consider the system in Problem 3.
Perform two subspace iterations with the starting vectors
⎡1 1 ⎤
X1 = ⎢⎢1 0 ⎥⎥
⎢⎣1 −1⎥⎦
That is, calculate X2 and X3, and hence the approximations to the exact eigenvalues and
eigenvectors (obtained in Problem 3).
2
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2.092 / 2.093 Finite Element Analysis of Solids and Fluids I
Fall 2009
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